Calculating dose‐averaged linear energy transfer in an analytical treatment planning system for carbon‐ion radiotherapy

Abstract Background Compelling evidence shows the association between the relative biological effectiveness (RBE) of carbon‐ion radiotherapy (CIRT) and the dose averaged linear energy transfer (LETd). However, the ability to calculate the LETd in commercially available treatment planning systems (TPS) is lacking. Purpose This study aims to develop a method of calculating the LETd of CIRT plans that could be robustly carried out in RayStation (V10B, Raysearch, Sweden). Methods The calculation used the fragment spectra in RayStation for the CIRT treatment planning. The dose‐weighted averaging procedure was supported by the microdosimetric kinetic model (MKM). The MKM‐based pencil beam dose engine (PBA, v4.2) for calculating RBE‐weighted doses was reformulated to become a LET‐weighted calculating engine. A separate module was then configured to inversely calculate the LETd from the absorbed dose of a plan and the associated fragment spectra. In this study, the ion and energy‐specific LET table in the LETd module was further matched with the values decoded from the baseline data of the Syngo TPS (V13C, Siemens, Germany). The LETd distributions of several monoenergetic and modulated beams were calculated and validated against the values derived from the Syngo TPS and the published data. Results The differences in LETds of the monoenergetic beams between the new method and the traditional method were within 3% in the entrance and Bragg‐peak regions. However, a larger difference was observed in the distal region. The results of the modulated beams were in good agreement with the works from the published literature. Conclusions The method presented herein reformulates the MKM dose engine in the RayStation TPS to inversely calculate LETds. The robustness and accuracy were demonstrated.


INTRODUCTION
Carbon-ion beam is a high linear energy transfer (LET) radiation with lower distal doses than the heavier ions but higher LET than the lighter ions. The variable LET of carbon-ion leads to variable RBEs. Therefore, CIRT requires an effective biophysical model for accurate quantification of RBEs throughout the entire range. 1 Local effect model (LEM) 2 and microdosimetric kinetic model (MKM) 2 are currently the most frequently used models in clinical practice. Although LEM and MKM use different approaches in calculating RBEs, [3][4][5] the dose averaged LET (LETd), taking into account the primary ions and the light-ion fragments, is the shared physical quality essential for determining the biological effectiveness. 6,7 Accordingly, ICRU-93 recommends reporting the dose-weighted LET for each treatment plan. 8 LETd also has been acknowledged as an indicator strongly correlated to the clinical outcome in cancer radiotherapy. [9][10][11] However, LETd calculation has not been widely available. Although dedicated Monte-Carlo (MC) simulations have been used in some institutions, 12,13 accessing a fast MC engine for routine LETd calculation is naturally challenging for most of the particle therapy practitioners. Developing an accurate and easy LETd calculation method on commercially available treatment planning system (TPS) platforms would therefore be advantageous.
Herein, a LETd calculation method for CIRT plans was established based on the RayStation TPS (V10B, Raysearch, Sweden). The function of the MKM-based pencil beam dose engine was reformulated from calculating RBE-weighted doses to obtaining the LETd. The accuracy of the method was subsequently validated by comparing its results with the published data and with Syngo, the TPS used in the routine clinical practice of our institution.

Procedural consideration
The kinetic energy is correlated with LET. Based on Equation (1), the LETd at the depth of x for a monoenergetic beam can be calculated: where i is ion type from Z = 1 to 6, j the kinetic energy of ion type i. LET ij (x), w ij , and d donate ion/energyspecific LET, relative weight, and the absorbed dose of the energy j of ion type i, respectively.
The RBE-weighted dose D RBE calculation in MKM is derived from the average number of lethal lesions in a nucleus, L n , as calculated according to Equation (2): where D Abs , D RBE , and S denote the ion absorbed dose, RBE-weighted dose, and the surviving fraction, a 0 and a r are the linear-quadratic (LQ) parameters of LET = 0 radiation and reference radiation. And is independent of radiation types. The critical parameter z * 1D in Equation (2) is represented as: where z * 1Di (x), w i , and d i (x) are z * 1D , the relative weight, and the absorbed dose of the ith beam. Subsequently, The objective here is to use the computational procedure of z * 1D (x) of Equation (3) in the TPS to obtain LETd(x) . This amounts to substituting the term z * 1Di (x) with LET ij (x) . With this substitution, the computation of D RBE would result in a fatigious quantity denoted as ξ LET , from which the LETd(x) can be recovered by the following:

TPS configuration and calculation
To implement the afore-described concept and technique, an independent non-clinical module, "LETd module", was created in the RayStation TPS for the sole purpose of calculating LETds. In this module, the z * 1Di (x) was replaced by LET ij (x), as such, the output of the RBE-weighted dose based on the analytic 2) would yield the factitious parameter ξ LET as described by Equation (5).
A two-step approach was performed for calculating the LETd for a given treatment plan: (i) the LETd module was applied to calculate the ξ LET based on the optimized D Abs ; (ii) the LETd was computed by Equation (6). Consistent with the algorithm used in the TPS, the calculation of LETd assumes water equivalence.

Validations
The Syngo system has been our primary TPS since the beginning of clinical operation in 2014, and was used to validate the proposed LETd inverse calculation method. Since the Syngo system does not provide LETd directly as part of the treatment plan output, an in-house software was developed to calculate the LETds of the Syngo plans using Equation (1) with the fragment spectra read out from the Syngo plans and the LET ij (x) in the Syngo beam data library.
To further consolidate the baseline beam data between the Syngo and RayStation for comparison purposes, the fragment-specific (Z = 1-6) LETds of a 418.74 MeV/u monoenergetic beam were generated from both Syngo and RayStation, and a scaling factor was introduced for each ion type. The LET ij (x) in the RayStation beam data library was subsequently adjusted by applying these scaling factors.

Monoenergetic beams
Three monoenergetic beams with 149.96, 258.75, and 348.51 MeV/u and a 3 mm ripple filter were used to generate dose distributions of 10 × 10 cm 2 field. The LETds from the Syngo plans were calculated using the in-house software and subsequently compared to the LETds from the RayStation plans using the new inverse method. The step sizes in the Bragg peak regions of the Syngo-LETds and the RayStation-LETds were < 1.0 mm and constantly 1.0 mm, respectively. The local differences between the two sets of LETds were  15 In this study, we generated the modulated plans using the same beam parameters of Inaniwa et al.'s study and then calculated the corresponding LETds, to evaluate the accuracy of our method.

Validation using monoenergetic beams
The LETds of 149.96 (a), 258.75 (b), 348.51 (c), and 418.74 (d) MeV/u are shown in Figure 2, with the difference (%) between the two methods of calculation overlaid.

DISCUSSIONS
An inverse approach for calculating the LETd using the RayStation TPS for CIRT plans was presented herein. This method can be implemented by creating a separate module using the same fragment spectra and the PBA dose engine in RayStation TPS. We employed this approach to calculate the LETds of several One of the benefits of CIRT is the higher RBE, attributed to its higher LETd. In-vivo studies show that high LETd can help overcome the radio-insensitive tumours under hypoxic conditions. 4,16,17 Furthermore, several recent studies have demonstrated the potential association of LETd of CIRT with tumour recurrence or normal tissue toxicities. [9][10][11] These suggest that clinical outcomes could be further improved by incorporating LETd in the planning optimization process. Although LETd can be derived through MC simulation, 18 measurements, 12 or analytic calculations, 19 evaluation or optimization of LETd in patients' treatment plans remains challenging. The first step towards this endeavour would be to make LETd distribution of a dose plan easily accessible. The described method allows the RayStation users to gain easy access to the LETd distribution of their plans and may allow further studies to correlate the clinical response with LETd in CIRT.
The currently available LETd calculations such as FRoG, 18 although quite powerful, requires additional beam commissioning, in particular, feeding a pencil beam dose engine with the Pre-MC-derived LETds of different energies. Our approach utilizes the exist-ing RBE-weighted dose engine without the need for additional commissioning data.
The sub-millimetre range deviations in Figure 2 are likely caused by interpolation of the beam parameters of arbitral energy from a set of discrete beam energies and fragment spectra. Furthermore, the LETd showed larger deviations in the tail of the LETds in depths. This can be explained by the fragment deviations of the ion types Z = 4 and 5 as shown in Figure 1.
Using PBA to inversely calculate the LETd with the presented algorithm inherently lacks the sensitivity to tissue variability and inhomogeneity (i.e., lung tissue), which should be taken into consideration when used in clinical settings.

CONCLUSIONS
A novel method of calculating LETd for CIRT plans was described. The accuracy and robustness were demonstrated. The process is based on the fragment spectra and PBA in the RayStation TPS, which could be easily implemented in any CIRT centre using RayStation TPS.

AU T H O R C O N T R I B U T I O N S
Jingfang Zhao and Xiaodong Wu designed the study. Weiwei Wu performed the study. Kambiz Shahnazi and Ping Li helped analysed the results. Weiwei Wu and Jingfang Zhao wrote the manuscript. All authors reviewed and approved the manuscript.